Predator-prey model with diffusion and indirect prey-taxis

We analyze predator-prey models in which the movement of predator searching for prey is the superposition of random dispersal and taxis directed toward the gradient of concentration of some chemical released by prey (e.g. pheromone), Model II, or released from damaged or injured prey due to predation (e.g. blood), Model I. The logistic O.D.E. describing the dynamics of prey population is coupled to a fully parabolic chemotaxis system describing the dispersion of chemoattractant and predator's behavior. Global-in-time solutions are proved in any space dimension and stability of homogeneous steady states is shown by linearization for a range of parameters. For space dimension N <= 2 the basin of attraction of such a steady state is characterized by means of nonlinear analysis under some structural assumptions. In contrast to Model II, Model I possesses spatially inhomogeneous steady states at least in the case N = 1.